Commutator of two subgroups is normal in join

From Groupprops
Jump to: navigation, search

Statement

Suppose H, K are subgroups of a group G. Then, the Commutator of two subgroups (?) [H,K] is a normal subgroup of the Join of subgroups (?) \langle H, K \rangle.

Facts used

  1. Subgroup normalizes its commutator with any subset: If H \le G is a subgroup and A is a subset of G, then H normalizes the following subgroup:

[A,H] = \langle [a,h] \mid a \in A, h \in H \rangle

Here, [a,h] = a^{-1}h^{-1}ah is the commutator of the two elements.

Proof

Given: Two subgroups H, K \le G.

To prove: [H, K ] \triangleleft \langle H, K \rangle.

Proof: By fact (1), H normalizes [K, H], which is the same as [H,K]. Also, K normalizes [H,K]. Thus, the normalizer of [H,K] in G contains both H and K, hence it contains \langle H, K \rangle, proving that [H,K] is normal in \langle H, K \rangle.